38
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Design of Six Sigma Supply Chains
D. Garg, Student Member, IEEE, Y. Narahari, Senior Member, IEEE, and N. Viswanadham, Fellow, IEEE
Abstract—Variability reduction and business-process synchronization are acknowledged as keys to achieving sharp and
timely deliveries in supply-chain networks. In this paper, we
introduce a new notion, which we call six sigma supply chains
to describe and quantify supply chains with sharp and timely
deliveries, and develop an innovative approach for designing such
networks. The approach developed in this paper is founded on
an intriguing connection between mechanical design tolerancing
and supply-chain lead-time compression. We show that the design
of six sigma supply chains can be formulated as a mathematical
programming problem, opening up a rich new framework for
studying supply-chain design optimization problems. To show
the efficacy of the notion and the design methodology, we focus
on a design optimization problem, which we call the inventory
optimization (IOPT) problem. Given a multistage supply-chain
network, the IOPT problem seeks to find optimal allocation of
lead time variabilities and inventories to individual stages, so as to
achieve required levels of delivery performance in a cost-effective
way. We formulate and solve the IOPT problem for a four-stage
make-to-order liquid petroleum gas supply chain. The solution
of the problem offers rich insights into inventory-service level
tradeoffs in supply-chain networks and proves the potential of the
new approach presented in this paper.
Note to Practitioners—This paper builds a bridge between
mechanical design tolerancing and supply-chain management.
In particular, the paper explores the use of statistical tolerancing techniques in achieving outstanding delivery performance
through variability reduction. Informally, a six sigma supply
chain is that which delivers products within a customer specified
delivery window, with at most 3.4 missed deliveries per million.
The innovations in this paper are the following: 1) to define two
performance metrics delivery probability and delivery sharpness
to describe the precision and accuracy of deliveries, in terms of
process capability indexes
,
, and
; 2) to formulate
the supply-chain design optimization problem using the process
capability indices; 3) to suggest an efficient solution procedure
for the design optimization problem. The paper presents the
case study of a two-echelon distribution network and using the
framework developed in the paper shows the role of inventory in
controlling lead time variability and achieving six sigma levels of
delivery performance.
Index Terms—Cycle time compression, delivery probability
(DP), delivery sharpness (DS), Motorola six sigma (MSS) quality,
process capability indexes (PCIs), process synchronization, six
sigma supply chains, supply-chain lead time, variability reduction.
Manuscript received September 5, 2002; revised June 16, 2003.
D. Garg and Y. Narahari are with the Department of Computer Science and
Automation, Indian Institute of Science, Bangalore 560 012, India (e-mail:
dgarg@csa.iisc.ernet.in; hari@csa.iisc.ernet.in).
N. Viswanadham is with the The Logistics Institute-Asia Pacific (a collaboration between the Georgia Institute of Technology and the National University
of Singapore), Singapore 119 260, and also with the Department of Mechanical and Production Engineering, National University of Singapore, Singapore
119 260 (e-mail: mpenv@nus.edu.sg).
Digital Object Identifier 10.1109/TASE.2004.829436
I. INTRODUCTION
S
UPPLY chains provide the backbone for manufacturing,
service, and E-business companies. The supply-chain
process is a complex, composite business process comprising
a hierarchy of different levels of value-delivering business
processes. Achieving superior delivery performance is the
primary objective of any industry supply chain. Quick and
timely deliveries entail high levels of synchronization among all
business processes from sourcing to delivery. This in turn calls
for variability reduction all along the supply chain. Variability
reduction and business-process synchronization are therefore
acknowledged as key to achieving superior levels of delivery
performance in supply-chain networks.
A. Motivation
Lead times of individual business processes and the variabilities in the lead times are key determinants of end-to-end delivery performance in supply-chain networks. When the number
of resources, operations, and organizations in a supply-chain
increases, variability destroys synchronization among the individual processes, leading to poor delivery performance. On the
other hand, by reducing variability all along the supply chain
in an intelligent way, proper synchronization can be achieved
among the constituent processes. This motivates us to explore
variability reduction as a means to achieving outstanding delivery performance. We approach this problem in an innovative
way by looking at a striking analogy from mechanical design
tolerancing.
Variability reduction is a key idea in the statistical tolerancing
approaches that are widely used in mechanical-design tolerancing [1]. A complex supply-chain network is much like a
complex electromechanical assembly. Each individual business
process in a given supply-chain process is analogous to an individual subassembly. Minimizing defective or out-of-date deliveries in supply chains can therefore be viewed as minimizing tolerancing defects in electromechanical assemblies. This analogy
provides the motivation and foundation for this paper.
In statistical-design tolerancing, process capability indexes
(PCIs) such as
and
[2], [3] provide an elegant
framework for describing the effects of variability. Best practices such as the Motorola six sigma (MSS) program [4] and
Taguchi methods [5] have been extensively used in design tolerancing problem solving. In this paper, we use these popular approaches in a unifying way to address variability reduction, synchronization, and delivery performance improvement in supplychain networks.
B. Contributions
The contribution of this paper is two fold. First, we recognize the key role of variability reduction and synchronization
1545-5955/04$20.00 © 2004 IEEE
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
in achieving superior delivery performance in supply-chain
networks and explore intriguing connections between statistical-design tolerancing and supply-chain lead time compression. Using this analogy, we introduce the notion of six sigma
supply chains. We show that the design of six sigma supply
chains can be expressed in a natural way as a mathematical
programming problem. This provides an appealing framework
for studying a rich variety of design optimization and tactical
decision making problems in the supply-chain context.
The second part of the paper proves the potential of the
proposed methodology by focusing on a specific design optimization problem which we call the inventory optimization
(IOPT) problem. We investigate this problem with the specific
objective of tying up design of six sigma supply chains with
supply-chain IOPT. Given a multistage supply-chain network,
the IOPT problem seeks to find optimal allocation of lead
time variabilities and inventories to individual stages, so as to
achieve required levels of delivery performance in a cost-effective way. The study uses a representative liquid petroleum
gas (LPG) supply-chain network, with four stages: supplier,
inbound logistics, manufacturer, and outbound logistics. The
results obtained are extremely useful for a supply-chain asset
manager to quantitatively assess inventory-service level trade
offs. For example, a supply-chain manager for the LPG supply
chain will be able to determine the optimal number of LPG
trucks to keep at the regional depot (RD) and the optimal
way of choosing logistics providers, so as to ensure six sigma
delivery of LPG trucks to destinations.
The supply chains for which the methodology discussed in
this paper can be categorized as “discrete event dynamical systems.” That is, the dynamics of the system is driven by occurrence of discrete events such as arrival of a customer order,
arrival of a truck at a logistics hub, dispatch of a truck from
a distribution center (DC), etc. All supply chains, regardless of
whether they are dealing with discrete parts or continuous processes can be modeled as a discrete event dynamical system.
The modeling, analysis, design, and optimization studies depend only on this “discrete event dynamics” of the system. The
continuous processes that may constitute individual subsystems
are modeled at an aggregate level, by representing their state
only at discrete epochs of time and through a stochastic representation of the lead time of the process. What is created here is
a “lead time model” which does not need to model the exact dynamics of the underlying production process. We only need to
model the starting epoch and completion epoch of the activities.
In our view, the concepts and approach developed in this
paper provide a framework in which a rich variety of supplychain design and tactical decision problems can be addressed.
C. Relevant Work
The subject matter of this paper falls in the intersection of
several topical areas of research. These include: 1) variability
reduction and lead time compression techniques for business
processes; 2) statistical-design tolerancing, and, in particular,
the MSS program; 3) IOPT in supply chains.
Cycle-time compression in business processes using variability reduction is the subject matter of a large number of
papers in the last decade. See, for example, the paper by
39
Narahari et al. [6], where variability reduction is applied to
subsystems to achieve lead time reduction of product development projects. Hopp and Spearman, in their book [7], have
brought out different ways in which variability reduction can
be used in compressing lead times of machining and other
business processes. Lead time compression in supply chains
(using variability reduction techniques) is the subject of several
recent papers, see, for example, Narahari et al. [8].
Statistical-design tolerancing is a mature subject in the design community. The key ideas in statistical-design tolerancing
which provide the core inputs to this paper are: 1) theory of
PCIs [2], [3], [9]; 2) tolerance analysis and tolerance synthesis
techniques [10], [11]; 3) the MSS program [4], [12]; 4) Taguchi
methods [5]; 5) design for tolerancing [1], [13].
IOPT in supply chains is the topic of numerous papers in the
past decade. Important ones of relevance here are those on multiechelon supply chains [14]–[16]. Variability reduction is a central theme in many of these papers. Recent work by Schwartz
and Weng [17] is particularly relevant here. This paper discusses
the joint effect of lead-time variability and demand uncertainty,
as well as the effect of “fair-shares” allocation, on safety stocks
in a four-link just-in-time (JIT) supply chain. Masters [14] develops an optimization model to determine near optimal stock
levels for multiechelon distribution inventories. His formulation is similar to what we have discussed as the IOPT model
in this paper, although his decision variables are different from
those identified here. Ettl et al. [15] develop an inventory-queue
model of a multiechelon supply chain with base stock policy followed at each store. Given the bill of materials, the nominal lead
times, the demand data, and the cost data, their model generates
the base stock level at each store that minimizes the overall inventory capital in the network and guarantees the customer service requirements. in several ways. The volume edited by Tayur
et al. [16] also contains several IOPT models in the supply-chain
context.
The salient feature of our model which makes it attractive
and distinguishes it from all the above discussed models, is the
notion of six sigma quality for the end-to-end delivery process.
Existing models in the literature consider either the availability
of the product to the customer as a criterion for customer service
level or probability of delivering the product to the customer
within a window as a measure of customer’s service level. Away
from these classical measurements of customer-service levels in
the IOPT problem, we propose a novel approach for customer
service level, namely accurate and precise deliveries, which is
the primary objective of any modern electronic supply chain.
The paper by Garg et al. [18] contains some of the core ideas
of this current paper and can be considered as a preliminary
version of the current paper. Another companion paper by the
authors [19] looks at a particular design-optimization problem
called the variance-pool allocation problem in a detailed way. In
contrast to [19], this current paper explores an IOPT problem in
multistage supply chains, after proposing a new notion, namely
six sigma supply chains.
D. Outline of the Paper
The paper is organized in following way. In Section II, we
review our work on supply-chain PCIs [18]–[21]. This review
40
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 1. Process variability and customer delivery window.
is provided for the sake of self-sufficiency of this paper. In this
, and
, and show their
review, we first define PCIs
relevance for modeling the delivery performance of the supplychain process. Next, we present the relationships among these
indexes. We define a new performance metric for delivery performance, namely delivery sharpness (DS), to supplement the
commonly used metric, delivery probability (DP). Finally, we
present an extension of the the MSS quality notion in order to
include DS. This prepares the ground for Section III.
In Section III, we first define the notion of six sigma supply
chains. Next, we show how the design of a six sigma supply
chain can be described in a natural way as a mathematical programming problem. We present several interesting design optimization and tactical decision making studies that can be carried
out using this framework.
Section IV is targeted to show the potential of the new approach developed in this paper. We consider a representative
four-stage supply-chain network for LPG distribution and formulate a design optimization problem, the IOPT problem, for
this network. Given required levels of DP and DS to be achieved,
the IOPT problem prescribes the optimal inventory level to be
maintained at a designated stage and the optimal way to distribute a pool of variance among lead times of individual stages.
Also, we describe a solution procedure for the IOPT problem
and present details of the solution approach for a realistic case
of the LPG supply chain. Finally, we touch upon numerical results and insights obtained.
Section V provides a summary of contributions of this
research and looks at several directions for future work in this
important area.
TABLE I
NOTATION USED IN DEFINITIONS OF PCIs
II. SUPPLY-CHAIN PCIs
As already stated, this section is a review of earlier work in
[18]–[20].
A. Introduction
, and
[2] are popular in the areas of
The PCIs
design tolerancing and statistical process control. Let us consider the situation depicted by Fig. 1 in order to describe the
idea of how capability of a process, where variability is an inherent effect, can be measured. The notation used in Fig. 1 is
listed in Table I. In Fig. 1, variability of the process is characterized by the probability density curve of the quality characteristic produced by the process, and customer specifications
are characterized by a delivery window which consists of tolerance and target value . Normal distribution is a popular and
because of its fundamental role in the
common choice for
theory of PCIs. The target value can be any value between
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
Fig. 2.
41
Process characterization.
and but we have assumed it as the mid point of two limits for
the sake of convenience.
Fig. 2 explains three possible geometries of the probability
density curve and customer delivery window when superimposed on each other. This curve is the crux behind the idea of
measuring the capability of a process.
B. Indexes
1) Index
is the cumulative distribution function of standard
where
normal distribution.
: Index
does not reflect the impact that
2) Index
shifting the process mean or target value has on a process’s
ability to produce a product within specification [9]. For this
index was developed.
is defined as follows:
reason, the
(3)
, and
: The PCI
is defined as
As assumed here, the target value is the mid point of , and
can be expressed in the following equivalent form:
(1)
where
tolerance
.
measures only the
potential of a process to produce acceptable products. It does
not bother about actual yield of the process where potential and
actual yield of any process are defined in the following manner.
Actual Yield: The probability of producing a part within
specification limits.
Potential: The probability of producing a part within specification limits, if process distribution is centered at the target
.
value i.e.,
It is easy to see [21] that the potential of the process is equal to
the area under the probability density curve taken from
to
when
and it can be expressed by the following
relation:
Potential
(2)
Here,
alone is not enough to measure actual yield of the
process. However, when used with , it can measure the actual
yield of the process. The formula for actual yield can be given
as below. A proof for this is provided in [21]
Actual Yield
(4)
: Actual yield of the process is related to the
3) Index
fraction of the total number of units produced by the process
which are defective, called as fraction defective. The fraction
defective is an indicator of process precision and it does not take
into account the accuracy of the process. In order to include the
notion of accuracy along with precision, we can use the index
[9] as follows:
(5)
The term
loss [3].
is known as the expected Taguchi
C. Relationship and Dependencies Among
, and
The following relations can be derived among
[21]:
(6)
42
Fig. 3.
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Typical variation of C
with respect to C for constant actual yield.
(7)
TABLE II
BOUNDS ON PCIs FOR Actual Yield
=
(8)
, and
have a tight coupling, in addition to the
PCIs
mutual relationship (6), (7), and (8), with process yield. It is easy
to show [21] that for a given value of actual yield (say), one
and
can define lower and upper bounds for the values of both
. We denote these lower and upper bounds by
,
and
, respectively. A crisp idea behind the intent of these
bounds is as follows:
• If the process’
is less than
, then, its
actual yield cannot be equal to , no matter how large
is.
is greater than or equal to
, then,
• If the process’
its actual yield cannot be less than , no matter how small
is.
is a little different. For any value of
• The case with
between
and
, it is possible to find a corresponding
such that the actual yield of the process is .
and
.
Table II summarizes such bounds on
Fig. 3 shows a typical variation of
with respect to
when the actual yield is constant.
Fig. 3 immediately leads to the following observations.
1) For a given pair
, the value of actual yield is
fixed.
2) However, for a given actual yield value, there exist infinite
such
pairs.
D. DP and DS
For every business process which is a part of a supply-chain
process, delivery time of product or service is an important
quality characteristic. Variability in lead time is inherent to
almost all the business processes, therefore, it will be apt to
apply the notion of PCIs to measure the delivery capability
or delivery quality of any business process. It is easy to see
from the relations presented in the Section II-C, that for a
given business process and for a given value of actual yield,
such that each one results
there exist infinite pairs
in the same actual yield. Similarly, by (8), for a given value
, there exists an infinite number of feasible
of
pairs. Nevertheless, it can be shown that for a given pair (actual
), there exists a unique feasible pair
.
yield
is sufficient to
This suggests that the 3-tuple
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
Fig. 4.
43
MSS quality in the presence of shifts and drifts in process mean.
measure the delivery quality of any business process in a given
can be substituted
supply chain. This 3-tuple
by the pair (actual yield,
) to measure the delivery quality.
In this paper, we use the term DP to refer to the actual yield of
a given process. It describes the precision of deliveries. We use
since it describes the accuracy of deliveries.
the term DS for
We use these two indexes to measure the quality of delivery
process in a given supply chain. Also, rather than expressing
the DP in terms of numerical values, we prefer to express it in
levels where
. We are motivated by the
terms of
MSS program [4] in using the idea of
levels. This will be
level corresponds to a
explained in the next section. Each
particular value of actual yield or equivalently a particular value
of number of defects per million opportunities.
E. MSS Quality Program: A Generalized View
The six sigma concept [4], [12] is a way to measure fractional
defectives in a lot. Six sigma quality is the benchmark for excellence of product and process quality. In this concept, a unique
level is attached with each value of number of defects per million
opportunities (npmo). The npmo is the probability, expressed on
, that a part is produced with quality charactera scale of
istic lying outside the specification limits. Here, it is assumed
that is normally distributed and the target value is the midand lower specification
point of upper specification limit
.
limit
It is not uncommon to the manufacturing processes that begins to drift away from the nominal value of engineering specifications as the machine tool begins to wear and other independent variables such as room temperature, material hardness,
etc., come into play. The shifting and drifting of process mean
is captured, in the MSS concept, by assuming a one sided mean
in the process mean. Also, it is assumed that
shift (bias) of
the variance of the shift in process mean is zero.
The idea behind fixing the value of levels in this case, is as
and
respecfollows. If and coincide with
and
(see Fig. 4),
tively, which are different from
then, the corresponding upper bound on yield will be assigned
level. Similar is the case with , , and others. Note that
the MSS program uses upper bound on yield (not actual yield)
in order to assign levels. However, we follow a slightly different approach. We say DP of the process is
iff actual yield
whereas according to the MSS
of the process is –
concept, six sigma quality is achieved when the the upper bound
.
of yield is –
As shown before, for a given
pair, the value of actual yield is fixed. But for a given actual yield value, there exist
pairs. Hence, DP can be completely deinfinite such
and
. However, there are numerous
termined by knowing
(in fact, infinitely many) ways in which we can choose the pair
to achieve a given value of DP. This leads to a generalized view of six sigma quality. MSS is a special case of this in
. In order to elaborate this idea let
which bias is fixed i.e.,
us start with the equation
Actual Yield
If we fix the value of the actual yield as in the above equation,
there will be two independent variables
, and the solution set will be unbounded. However, we have earlier shown that
, and
are bounded within cerfor a given actual yield
tain range. Therefore, the solution is bounded by
. If we substitute
–
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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 5. The curves and C
curves on C
0C
plane.
and plot a graph, then, all the points lying on the curve will give
pairs that result in
quality level. This equation can
be generalized for any
level by expressing in terms of .
It is easy to see from Fig. 4 that the upper bound in the MSS
level is
. Equating this to the actual
program for
quality
yield of the process we get the following equation for
curve on the
plane, we get
Some of these curves are plotted in Fig. 5. We can proceed
one step further by looking at the connection between DP and
DS in the light of our generalized notion of six sigma quality.
For this, we consider the plots of quality levels on
plane and then see how
varies on the same plot. To see
, and
this, we use the identity relation (8) among
and plot this relation for a constant value of
(say
). The
plot comes out to be a section of a hyperbola. From a process
design point of view, it can be said that for a desired level of
) and DP (i.e.,
), this curve provides a set
DS (i.e.,
of 3-tuples
which all satisfy these two requirements. The designer has to decide which one of the triples to
choose depending upon the requirements. Fig. 5 shows some
curves on the
plane.
III. SIX SIGMA SUPPLY CHAINS
TABLE III
SAMPLE VALUES OF 3-TUPLES (C ; C ; C ) WHICH
ACHIEVE SIX SIGMA DELIVERY PERFORMANCE
which, given the customer specified window and the target delivery date, results in defective deliveries (i.e., DP) not more
that guarantee an acthan 3.4 ppm. All triples
) would correspond
tual yield of at least 3.4 ppm (or
to a six sigma supply chain.
Table III provides sample values of PCIs that achieve six
sigma delivery performance. It is important to note that in order
, the DS needs to assume appropriately
to achieve
high value. In a given setting, however, there may be a need for
extremely sharp deliveries (highly accurate deliveries) implying
index is required to be very high. This can be specthat the
ified as an additional requirement of the designer.
A. Notion of Six Sigma Supply Chains
B. Design of Six Sigma Supply Chains
Motivated by the discussion in the previous section, we seek
to define the notion of six sigma supply chains, to describe a
supply chain with superior delivery performance. We define a
six sigma supply chain as a network of supply-chain elements
We can say that the design objective in supply-chain networks
is to deliver finished products to the customers within a time as
close to the target delivery date as possible, with as few defective deliveries as possible at the minimum cost. To give an idea
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
of how the design problem of a complex supply-chain network
can be formulated, let us consider a supply chain with business
processes such that each of them contributes to the order-to-debe the cycle
livery cycle of customer desired products. Let
is a contime of process . It is realistic to assume that each
and standard deviation
tinuous random variable with mean
. The order-to-delivery time can then be considered as a
deterministic function of ’s
If we assume that the cost of delivering the products depends
only on the first two moments of these random variables, the
total cost of the process can be described as
where is some deterministic function.
The customer specifies a lower specification limit , an upper
specification limit , and a target value for this order-to-delivery lead time. With respect to this customer specification, we
so as to
are required to choose the parameters of
minimize the total cost involved in reaching the products to the
customers, achieving a six sigma level of delivery performance.
Thus, the design problem can be stated as the following mathematical programming problem:
Minimize
subject to
DS for order-to-delivery time
DP for order-to-delivery time
45
2) Concrete Design Problems:
• due date setting;
• choice of customers;
• inventory allocation;
• capacity planning;
• vendor selection;
• choice of logistics modes, logistics providers;
• choice of manufacturing control policies.
These problems can arise at any level of the hierarchical design.
Thus, in order to develop a complete suite for designing a complex supply-chain network for six sigma delivery performance
through the hierarchical design scheme, we need to address all
such subproblems beforehand. In the Section IV, we consider
one such subproblem, optimal allocation of inventory in a multistage six sigma supply chain, and develop a methodology for
this problem.
IV. INVENTORY OPTIMIZATION IN A
MULTISTAGE SUPPLY CHAIN
In this section, we describe a representative supply-chain
example for LPG, with four stages: supplier (refinery), inbound
logistics, manufacturer [regional depot (RG) for LPG], and
outbound logistics [17]. We formulate the six sigma design
problem, based on the concepts developed in earlier sections,
for this supply chain. Then we show how one can allocate
variabilities to lead times of individual stages so as to achieve
six sigma delivery performance. We also show how to compute
the optimal inventory to be maintained at the RD to support six
sigma delivery performance.
A. Four-Stage Supply-Chain Model With Demand and Lead
Time Uncertainty
where
is a required lower bound on DS. The objective
function of this formulation captures the total cost involved
in taking the product to the customer, going through the individual business processes. We have assumed that this cost is
determined by the first two moments of lead times of the individual business processes. One can define in a more general
way if necessary. The decision variables in this formulation are
means and/or standard deviations of individual processes. The
constraints of this formulation guarantee a minimum level of
is the minimum level of DS required) and at least a
DS (
six sigma level of DP.
While solving the design problem, an important step is to
express the constraints in terms of the decision variables. This
will be elaborated upon in the next section.
C. Representative Design Problems
Depending on the nature of the objective function and
decision variables chosen, the six sigma supply-chain design
problem assumes interesting forms. We consider some problems below under two categories: 1) generic design problems;
2) concrete design problems.
1) Generic Design Problems:
• optimal allocation of process means;
• optimal allocation of process variances;
• optimal allocation of customer windows.
1) Model Description: Consider
geographically dispersed DCs supplying retailer demand for some product as
shown in Fig. 6. The product belongs to a category which does
not make it profitable for the DC to maintain any inventory. An
immediate example is a distributor who supplies trucks laden
with bottled LPG cylinders (call these as LPG trucks or finished
product now onward) to retail outlets and industrial customers.
In a situation like this, as soon as a demand for an LPG truck
arrives at any DC, the DC immediately places an order for one
unit of product (in this case, an LPG truck) to a major RD. The
RD maintains an inventory of LPG trucks and after receiving
the order, if on-hand inventory of LPG trucks is positive, then,
an LPG truck is sent to the DC via outbound logistics. On the
other hand, if the on-hand inventory is zero, the order gets
backordered at the RD. At the RD, the processing involves
unloading the LPG from LPG tankers into LPG reservoirs,
filling the LPG into cylinders, bottling the cylinders and finally
loading the cylinders onto trucks.
The inventory at RD is replenished as follows. The RD starts
with on-hand inventory and every time an order is received,
it places an order to the supplier for one LPG tanker (called the
semifinished product now onward) which is sufficient to produce one LPG truck. In this case, the supplier corresponds to a
refinery which will produce LPG tankers. In the literature such
model [22] with
a replenishment model is known as the
46
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 6. Four-link linear supply-chain model.
. In such a model, the inventory position (on-hand plus
on-order minus backorders) is always constant and is equal to .
It is assumed that raw material (crude oil or naphtha) required
for producing an LPG tanker is always available with the refinery, but the refinery needs to do some processing of this raw
material to transform it to LPG and load it onto a tanker. Therefore, as soon as the refinery receives an order from the RD, it
starts processing the raw material and sends an LPG tanker via
inbound logistics to the RD.
The LPG supply-chain network is a typical example of a multiechelon supply-chain network. The four stages could be considered generically as procurement, inbound logistics, manufacturing, and outbound logistics, as described below (descriptions
in parentheses corresponds to the LPG example).
1) procurement or supplier (refinery);
2) inbound logistics (transportation of LPG tankers from refinery to RD);
3) manufacturing (RD);
4) outbound logistics (customer order processing and transportation of LPG trucks from RD to a DC).
The DCs in the LPG example correspond to the end user in
the general setting of a four-stage supply-chain network. This
example emphasizes the distribution aspect of the supply chain.
In the next section, we articulate all the assumptions we have
made regarding behavioral and operational characteristics of
this model. We believe that these assumptions are reasonable
enough to make the model realistic and yet tractable, preserving
all important insights of the problem.
2) Assumptions:
1) A customer places order for only one unit of finished
product at a time to the manufacturer.
2) The orders arrive at the manufacturer in Poisson fashion
from each customer. The Poisson arrival streams of orders
are independent across the customers.
3) Each customer specifies a delivery window while placing
an order. This window is assumed to be the same for all
the customers. Also, in this window the date which customer targets for delivery of the item has equal offset from
the upper specification limit and lower specification limit.
4) If the item is not on-hand with the manufacturer, then, the
customer’s order gets backordered there. All such backorders are fulfilled in a first in first out (FIFO) manner by
the manufacturer.
5) Lead time for an item at each stage of the supply chain is
a normal random variable. Lead times of the four stages
are mutually independent. As soon as a supplier receives
an order from the manufacturer, processing commences
on the corresponding raw material.
6) Inbound and outbound logistics facilities are always available. Therefore, as soon as an item finishes its processing
at the supplier, its shipment starts via the inbound logistics. Similarly, as soon as an order of a customer is received by the manufacturer, the shipment of an item, if
available, commences using the outbound logistics. Otherwise the shipment commences as soon as it becomes
available at the manufacturing node (following an FIFO
policy). Inbound logistics lead times are independ identically distributed (i.i.d.) random variables and outbound
logistics lead times are also i.i.d. random variables.
7) An item that arrives from the supplier does not wait in
queue at the manufacturer for getting processed (there is
no queuing before the manufacturing stage). Processing
times for items at the manufacturing node are i.i.d.
random variables.
8) The processing cost per item at each stage depends only
on the mean and variance of lead time of the stage.
9) Costs related to maintenance of inventory at the manufacturing node are fixed. Such costs include order placing
cost, inventory carrying cost, cost of raw material of an
item, and fixed cost against backorder of an item. However, the variable cost of the backorder is a function of
time for which the order is backordered.
Note that in this example, the following hold.
• We have modeled the dynamics of each of the four individual stages by a corresponding normally distributed lead
time random variable.
• We have assumed only one stock, at the RD. We could
have assumed inventories at the other three stages, but that
would not add much to the insights sought to be illustrated
here.
• We have modeled the demand process.
• We have modeled stockouts, backordering, and inventory
replenishments corresponding to the inventory at the RD.
3) System Parameters: This section presents the notation
used for various system parameters.
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
Lead Time Parameters:
Procurement lead time.
Inbound logistics lead time.
Manufacturing lead time.
Outbound logistics lead time.
Time elapsed between placement of an
order by manufacturer and receipt of the
order from supplier.
Time elapsed between placement of an
order by manufacturer and completion of
the order manufacturer.
End-to-end lead time of customer’s order.
An upper bound on .
Mean and variance of
.
Mean and variance of .
Mean and variance of .
Mean and variance of .
Demand Process Parameters:
Order arrival rate from th customer (item/year).
, Poisson arrival rate of orders at the manufacturer.
Inventory level at the manufacturing node.
Reorder quantity of the manufacturer.
Stockout probability at the manufacturing node.
Average number of backorders per unit time at the
manufacturing node (item/time).
Expected number of backorders with the manufacturer at arbitrary time (item).
Expected number of on-hand inventory with the manufacturer at arbitrary time (item).
Steady-state probability that the manufacturer has a
net inventory equal to .
.
.
Cost Parameters:
, Procurement cost ($/item).
, Inbound logistics cost ($/item).
, Manufacturing cost ($/item).
, Outbound logistics cost ($/item).
Order placing cost for manufacturer ($/order).
Fixed part of backorder cost ($/item).
Variable part of backorder cost ($/item-time).
Inventory carrying cost ($/time-invested).
Cost of raw material($/item).
Capital tied up with each item ready to be
shipped via outbound logistics ($/item).
Delivery Quality Parameters:
Supply-chain PCIs for end-to-end lead time
of customer order.
Delivery window specified by customer.
Upper limit of delivery window.
Lower limit of delivery window.
Bias for .
Bias for .
.
.
47
B. System Analysis
1) Lead Time Analysis of Delivery Process: In this section,
we study the dynamics of material flow in the four-stage supply
chain described earlier. This is a classic example of a discrete
event dynamical system (a class of systems where the dynamics
is completely determined by the occurrence of events at discrete
epochs). The various events of the supply chain, that are of our
interest, are illustrated in Fig. 7.
In this section, we wish to present a couple of observations
about the lead time of the delivery process. These observations
will give a sense of end-to-end lead time and will also serve
as building block while formulating the IOPT problem. Some
of these observations are described in the form of Lemma 1.
A few of them, that follow immediately from system dynamics,
are stated below without any explanation.
1) End-to-end lead time experienced by manufacturer after
) is given by
placing an order to supplier (i.e.,
2) The time taken to get finished product ready, after manufacturer places the corresponding order for semifinished
product to supplier (i.e., ) is given by
The lemma below provides an upper bound on end-to-end lead
time experienced by an end user.
Lemma 1: The upper bound on end-to-end lead time
experienced by an end customer, in a four-stage (LPG) supply
chain described above, is given by
where
is the stockout probability at the manufacturer (RD).
Proof: Recall that the arrival of an end customer order
triggers placement of an order request by the manufacturer to the
supplier. Also, the item is shipped to the end customer immediately if it is available in stock at the manufacturer, otherwise the
order gets backordered at the manufacturer (see Fig. 7). In the
first case, the lead time experienced by the end customer will be
. For the second
the same as the outbound logistics time i.e.,
case, assume that the customer order is the th backorder at the
. By virtue of the invenmanufacturer where
tory replenishment policy followed by the manufacturer, there
outstanding orders of semifinished products imwill be
mediately after arrival of this backorder. Remember that it is an
, and
are
underlying assumption of the model that
independent across items also. Hence,
is independent across
orders. It is a direct consequence of this result
all these
that the orders placed by the manufacturer can cross each other
which means that a product for which supply-chain activities
were started later may be ready in finished form earlier than the
product for which activities were started earlier. A comprehensive idea of this phenomenon is presented in [22].
48
Fig. 7.
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Event diagram for four-stage supply-chain model.
In view of the crossing of finished products at the manufacturer, it is easy to see that the finished product which is allocated
to some backorder may not be the one that results from the corresponding order placed by the manufacturer to the supplier on
arrival of this backorder. If it were so, the time taken to serve a
backorder by the manufacturer would have been no more than
. Thus, due to:
1) crossing of orders at manufacturer;
2) assumption of indistinguishable products;
3) FIFO policy for serving the backorders;
. Hence it can
this time is definitely less than
be said that random variable
gives an
upper bound on the time taken to get the finished product by
end customer in the second case.
Considering the first and second case together and using
theory of total probability, it is easy to establish following
expression for upper bound on lead time experienced by end
customer
an upper bound on it. It is interesting to observe that as the variability in lead time or demand process reduces, the likelihood
closer to . Forof crossover reduces which in turn brings
is a monomally, it can be said that the difference
tonically increasing function of crossover probability .
2) Analysis of Inventory at the Manufacturing Node: Obpolicy, with
,
serve that the manufacturer follows a
for replenishing the finished product inventory (such a policy is
also called as the base stock policy in inventory management
literature). There is a well-known theorem (after Tackács 1956)
models with
which is reproduced below.
[22] for
policy with
be followed
Theorem 1: Let the
for controlling the inventory of a given item at a single location
where the demand is Poisson distributed with rate , and let the
replenishment lead times be nonnegative independent random
and mean .
variables (i.e., orders can cross) with density
The steady-state probability of having net inventory (on hand
inventory minus backorders) by such a system can be given by
It is now clear that in the absence of crossover, the end-to-end
lead time for the second case will be equal to
.
Hence, will represent the end-to-end lead time rather than
In other words, the steady-state probabilities are independent of
the nature of the replenishment lead time distribution if the lead
times are nonnegative and independent.
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
49
In the context of the four-stage supply chain, the replenishwhich has already been
ment lead time for manufacturer is
shown to be independent over finished products (i.e., finished
is a normal random
products can cross each other). However,
variable which is not nonnegative. Therefore, the above theorem
cannot be applied to finished product inventory directly.
Nevertheless, it is safe to assume that the probability of
, and
taking negative values is small enough that
without
the above theorem can be applied for lead time
then, the negative
significant error. For example, if
is no more than
area of the probability density function of
, which can be ignored for all practical purposes and
can be assumed as virtually nonnegative. In view of this
argument, the steady-state probability of having a net inventory
of finished products with the manufacturer can be given
as follows:
(11)
the specified levels of DP and DS are achieved for the end customer lead time, in the steady-state condition, in a cost effective
manner. We call this problem as the IOPT problem in six sigma
supply chains.
results
It is easy to see that an increase in the value of
in high inventory carrying cost, and improved quality of deliveries. Similarly, variance reduction of lead time at any stage(s)
of the supply-chain results in a high processing cost and improved quality of deliveries. This means that a specified level
of quality for the delivery process can be achieved either by increasing the value of or by reducing the variance of lead time
for one or more stages or both. The problem here is to determine a judicious balance between these two such that the cost
is minimized.
or
, there is a slight
Depending upon whether
change in the formulation of the problem. Therefore, we formu(which we
late two separate IOPT problems for the cases
(which we call “with zero stock”).
call “with stock”) and
Since in both cases, we use a make-to-order policy to pull the
products, we can more completely describe these two policies
as make-to-order-with-stock (MTOS) policy and make-to-orderwith-zero-stock (MTOZS) policy, respectively. The input parameters and decision variables are the same for both MTOS Policy
and MTOZS Policy. However, the objective function as well as
the constraints are different for these two policies. The input parameters, decision variables, objective function, and constraints
in the IOPT problem are as follows.
1) Input Parameters: The input parameters to the IOPT
,
problem are: mean of random variable , for
arrival rate for customer orders, customer delivery window
, desired levels of DP
and DS
for customer
lead time, and coefficients of the first three terms in Taylor
for
.
series expansion of processing costs
is a function of and
It is assumed in Section IV.A2 that
. But ’s are known in the IOPT problem. Therefore,
is
only. Hence, the first three terms in the
now a function of
Taylor series expansion of
can be given as follows:
(12)
(15)
serves in deriving an important conThe expression for
clusion about upper bound on end-to-end lead time for customer (i.e., ) which is described (without proof) in the form
of Lemma 2.
, and , the upper bound
Lemma 2: For a fixed value of
on end-to-end lead time experienced by an end customer (i.e.,
) is a normal random variable with mean and variance
given by
The coefficients
, and
do not have an immediate
physical significance and will have to be determined by the
supply-chain designer using data available about the supplychain business processes.
2) Decision Variables: The decision variables in IOPT are
,
standard deviation of each individual stage
and the inventory position .
3) Objective Function:
MTOS Policy: We identify the following costs as significant costs for this policy:
$/year;
• average annual processing cost for supplier
$/year;
• average annual inbound logistics cost
$/year;
• average annual manufacturing cost
$/year;
• average annual outbound logistics cost
• average annual order placing cost
$/year;
$/year;
• average annual backorder cost
$/year;
• average annual inventory carrying cost
$/year.
• average annual cost of raw material
Now, it is no more difficult [22] to derive the expressions for the
, the average number of backorders
stockout probability
, the expected number of backorders at any
per unit time
random instant
, and the expected number of on-hand inven. These expressions are listed as
tory at any random instant
(9)
(10)
(13)
(14)
where
is given by (9).
C. Formulation of IOPT
The objective of the study here is to find out how variability
should be allocated to the lead times of the individual stages and
what should be the optimal value of inventory level , such that
50
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
The sum of all the above mentioned costs gives the total average
annual operating cost, say COST. This comes out to be
(16)
It is easy to see that capital tied up with each unit of finished
good, ready to be shipped at manufacturer, includes: raw material cost, total processing cost excluding outbound logistics cost,
and order placing cost against this finished good. This comes out
to be
It is required to express these constraints in terms of decision
variables ’s before we can formulate optimization problem.
with variances of inRecall, Lemma 2 relates the variance
dividual stages, for a given value of (note that and
are
can be expressed
known here). Thus, for a given value of
and
of
in the following manner:
in terms of
(24)
where , the tolerance of customer delivery window, is a known
parameter in the IOPT problem and is given as follows:
(25)
(17)
Substituting the value of
tion, we get
This results in
(18)
Observe that, for a particular value of
are all known constants and hence they can be combined into one single constant and the equation reduces to
(19)
where
to express
In the above equation
are all known parameters. Also,
, according to (9), depends only on
, and .
is a known parameter. The
Therefore, for a given value of
and
. Substituting
only unknown quantities in (24) are
the value of (24) in (14) we get the following relation which is
the crux of the problem of converting constraints in terms of decision variables, for a given value of
. If we use (15)
then the above expression becomes
(20)
MTOZS Policy: This policy is a special case of the previous one in which the inventory carrying cost need not to be
considered. The processing costs for each stage in this policy
are the same as those of the MTOS Policy because these costs
have no relation to the finished product inventory at the manufacturer. Also, raw material cost and order placing cost are the
same as in the previous one. However, the expression for the cost
of backorders is a little different. Under this policy, every order
of the customer which arrives at the manufacturer does not find
the finished product and therefore gets backordered there. This
so the values of and become and
,
implies
respectively. In view of this, the annual backorder cost becomes
. Summing up all the significant costs, we get the
following expression for COST in this policy:
(21)
4) Constraints: Recall that is an upper bound on end customer lead time . Hence, if we specify the constraints which
assure to attain the specified levels of DP and DS for , it will
automatically imply that attains the same or even better levels
of DP and DS than the specified. These constraints can be given
as follows:
DS for
(22)
DP for
(23)
, from Lemma 2 in the above rela-
(26)
The above relation is the crux behind expressing the constraints
(22), and (23) in terms of decision variables. The idea is like
this: choose the values of index
and
for in such a way
that desired level of DP and DS are ensured. Use these values in
above relation. This idea is elaborated in the next section. Under
the MTOZS policy, the above expression remains the same ex.
cept that
D. Solution of IOPT
Observe that the objective function is a function of ’s
and
which are all functions of . Theoretically, can
take any value from set of natural numbers and ’s can take any
positive real value. It makes the optimization problem a mixed
integer nonlinear optimization problem. Fortunately, cannot
take any arbitrarily large value. For example, a seasoned asset
manager who is engaged in managing the inventory can tell by
his experience that can never exceed a certain value. Also,
often times, there is a constraint on storage space, or capital
tied up with inventory, etc. which further limits the value of .
Therefore, a good way of solving the IOPT problem would be
the following.
1) Fix a value for
and solve the resulting subproblem
to determine optimal values of ’s to achieve the optimal COST for that value of . This requires a careful
study and interpretation of the constraints to determine
and
for a given value of . This is
the values of
discussed in the next subsection.
.
2) Repeat Step 1 for all possible values of
. The case
is a bit dif3) Repeat Step 1 for
ferent from that of
since it leads to a subtly dif-
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
Fig. 8.
Possible geometric shapes of feasible region for C
51
and C
of L .
ferent objective function and subtly different constraints
(for details, see [21]).
4) Determine the minimum among all such optimal upper
bounds on COST computed above. The corresponding
will give the optimal inventory level to be maintained and
the corresponding ’s will give the optimal variabilities
to be assigned to individual lead times.
Thus, the complexity of the IOPT problem will be equal to
where
is the upper bound
on inventory and is the complexity for solving the optimization problem for a fixed value of . Thus, the first term of this
expression gives the complexity involved in executing the Steps
1 through 3 of the above algorithm and second term represents
the complexity of the Step 4.
(23) and therefore, can be used as a design point in (26). The
concern here is which point should be selected as design point.
Before we investigate further in this direction, let us consider a
pair in the form of
few interesting facts about such a
two Lemmas (3, 4). These two lemmas are proved in [19] and
we reproduce the proofs here for the sake of self-sufficiency.
is bounded
Lemma 3: For given values of and , DS of
above by
Proof: For a given value of and
, and
of the
need to satisfy the following relation (see (26)):
process
(27)
E. Determining
and
for a Given Value of
The unknown pair
in (26) is chosen in a way that
it satisfies both the constraints (22) and (23). The idea behind
getting such a pair is as follows. The relation (26) forces the depair to lie on the line
in the
sired
plane. Also, it is easy to see that the constraint (22)
forces the desired pair to lie on or above the curve
in
plane. Similarly, constraint (23) forces it to lie on
curve in the same plane. All these result in a
or above the
feasible region in the
plane. Fig. 8 shows all possible
geometries for such a feasible region, depending upon the relcurve and the
curve. From
ative position of
Fig. 8, it is clear that the feasible region in each case is the part
, denoted by EP, which intersects
of the line
the shaded region. For the sake of clarity, we have shown the
line OP only in Case 1. In all other cases it is understood. Each
point of the feasible region satisfies both constraints (22) and
If we take any point on this line, it represents a unique combination of
, and
. Hence, if we choose this point as
gets fixed. Now, consider the foldesign point, the DS for
curve on the
plane:
lowing equation for a typical
It can be verified that this equation represents a hyperbola. It
is quite possible that the line given by (27) becomes an asymptote of such a hyperbola. Such a hyperbola is the curve of
because it is clear from the geometry of the figure that this line
curve which is greater than
.
cannot intersect any other
value (or DS) higher
Hence, it is not possible to achieve the
than
for process , under the given and .
curve
It is easy to show that the slope of asymptotes of
. Equating these to the slope of the line (27)
is
we get the required expression for
.
52
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 9. Optimal inventory level R for DP = 3 and DS = 0:7.
Lemma 4: For given values of and , a unique value of DP
gets fixed automatically for
whenever it is attempted to fix
and also vice versa. Moreover, these DP and DS have
DS for
a positive correlation.
pair is chosen
Proof: Earlier, we said that the
for
in a way that apart from satisfying both the constraints
(22) and (23), the pair must lie on the line (27).
curve and a unique
It is easy to verify that a unique
curve pass through a unique point of the line (27). These
values and
values are final DP and DS, respectively, which
if this particular point is chosen as design
are achieved for
point. Hence, it can be concluded that once a value is chosen for
DP of , it will automatically decide the corresponding value
of DS and also vice versa. To prove the other statement of the
to point
lemma, observe that as we move from point
on the line (27), the values of both
curve and
curve which pass through that point increase. Therefore, DP
increases (or decreases) as DS increases (or decreases) for given
values of and .
The implication of Lemma 1 is as follows. If the desired
is greater than
for given values of and , then, the
problem is infeasible. In such a situation, we need not proceed
any further. Lemma 2 also has a key implication on the problem
and
for . According to Lemma
of fixing the values of
get fixed immediately as soon as a feasible
2, DP and DS of
point from line (27) is chosen as design point. It is easy to see
plane is unique on its own bethat each point on the
cause it has a unique combination of DP and DS. Therefore, it
is quite possible that the point which we have chosen results in
either higher DP or higher DS than required for the end-to-end
delivery process. Hence, it can be claimed that it is not always
from design are exactly
true that the DP and DS obtained for
the same as given in constraints (22) and (23).
In view of the above findings, the problem of fixing the values
and
can be addressed as follows. First step toward
of
this is to test the feasibility of the problem through Lemma 1.
If the problem turns out to be feasible then each point in the
feasible region is allowed to be chosen as design point. However, depending upon the point which is chosen as design point,
(which we get out of solving the optimization
the final cost
problem) may vary. At this point, we cannot say which feasible
point will result in minimum cost. Hence, the problem is handled in an indirect manner. The proposed scheme is like this.
First, solve the optimization problem without any constraint and
for . It will result in a global minget the optimal variance
to get
and
for
imum cost. Now, use this variance
which result in minimum cost. If the point
falls
in the feasible region, then, this point is used as a design point
; otherwise, the point where the line OP enters the
pair. The
shaded region is taken as the final desired
reason behind choosing point as design point is as follows.
The values DP and DS which result from point are the minimum possible values satisfying both the constraints (22) and
(23). If we choose any other feasible point, then, even though
will satisfy the constraints (22)
the resulting DP and DS for
and (23), yet their values will be a bit high and this will lead to
higher cost. In this way, we convert the constraints in terms of
decision variables for a given value of .
An important point to note here is that (14) holds true only
is negligibly small. In order for this
when the negative area of
condition to hold, it is necessary that the following constraint
must also be satisfied along with constraint (26):
(28)
(29)
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
53
The following equality and inequality constraints are now ready
for the IOPT problem:
truck
truck
(30)
truck year
year invested
truck
(31)
(32)
The MTOZS policy is a special case of the MTOS Policy with
. Under this policy,
. Therefore, the constraint
. However, the constraint
(30) remains the same except
irrespec(31) is no more needed because in this case
tive of whether
is nonnegative or not. Following are the two
insights about these constraints.
• Because the formulated constraints ensure the provided
levels of DP and DS for
rather than for , the minimum cost which we get after solving this problem is actually greater than or equal to what is actually required to
achieve the specified levels of DP and DS for . In other
words, it can be said that the COST which we get here is
an optimal upper bound on the COST for achieving specified levels of DP and DS on .
• If one looks at the purpose of constraint (31), then it is easy
without which
to see that it enforces nonnegativity of
. We imposed
it is not possible to use formula (9) for
to ensure this nonnegativity and becondition
cause of that only we got this constraint. We could have
but in that case error involved
as well chosen
with the help of formula (9) would
in computing the
have been higher. The maximum error that we can tolerate
in the results will determine whether or not this constraint
is needed in the ultimate analysis. If the constraint is removed, the optimization problem will turn out to have only
equality constraints.
for the given , we will thus
Having obtained a pair
be required to solve a nonlinear optimization problem with
equality constraints. The Lagrange multiplier method can be
used here.
4) Delivery Quality Parameters:
days;
days
For the sake of numerical experimentations, we consider the following four different sets of constraints and solve the problem
under each case:
and
for
(see Fig. 9);
1)
2)
and
for
(see Fig. 10);
and
for
(see Fig. 11);
3)
and
for
(see Fig. 12).
4)
Assume that it is not possible for the RD to keep more than
40 LPG trucks ready at any given point of time.
We first describe Step 1 of the procedure to solve IOPT, discussed in the last section, for this numerical example. Let us
and
to work with.
choose constraint set
Step 2 can be carried out in the same manner for all the other
values of . Step 3 and Step 4 are also trivial. The same procedure can be repeated for other constraints sets also.
. We first compute the folTo start with, let us fix
lowing parameters for the given numerical values:
6 days
trucks year
trucks
trucks
days
Substitution of these values in (20) and (30) results in the following optimization problem:
Minimize
F. Solution of IOPT for a Specific Instance
Let us consider the LPG supply chain once again and study
the problem in a realistic setting. We have chosen the following
values for typical known parameters of the IOPT problem in the
context of the LPG supply chain. The values we have chosen.
1) Lead Time Parameters:
day,
days,
days, and
days.
trucks/year.
2) Demand Process Parameters:
3) Cost Parameters: These parameters have been chosen so
as to capture the negative correlation between cost and mean
lead time and between cost and variability of lead time
truck
truck
truck
(33)
where
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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 10. Optimal inventory level R for DP = 4 and DS = 0:8.
subject to
The constants
can be determined with the help of Taylor
series expansion of the cost functions . We have expanded
and used the corresponding
all the cost functions at
. The immediate problem is to
coefficients as the constants
and
. As a first step toward this, it is
find out values of
required to check the feasibility of the problem as per guidelines
provided in Lemma 1. Note the upper bound on DS for this
case is
Hence, as far as feasibility is concerned, there is no problem because all the desired values of DS are within permissible range.
and
that results in
As a next step we find out the pair
global minimum and test whether it belongs to the feasible region or not. For this, let us assume that
. It immediately follows from this definition of that
where is a nonempty open
convex set. To test the convexity of objective function , we
and Hessian matrix
for
compute gradient vector
at point
. Observe that the
function
gradient vector and Hessian exist for each
. It directly
is twice differentiable over . Morefollows that function
over, the Hessian is independent of . Therefore, it is sufficient
that we test the positive definiteness (PD) or positive semidefi-
niteness (PSD) of the Hessian at any one point of instead of
testing it all over .
It is easy to see that all the diagonal elements of the Hesare positive. Theresian are positive real numbers because
fore, the Hessian is PD and function is strictly convex which
implies that a local optimal solution of unconstrained problem
is the unique global optimal solution. This can be obtained by
to 0. For the present numerical example it reequating
sults in
days. These
can be
used to find out which comes out to be 2.665 44 days. Indexes
and
can be computed by using . For the present exand
.
ample, these indexes are
These
and
can further be utilized to determine the value
of DP and DS at the global minimum point which come out to
and
respectively. These quality levels are
be
and
as design
more than what is desired. Hence, we use
values. If these quality levels come out to be less than specified
in the constraints, it is required to use the scheme suggested earlier in Section IV.C.4. Substituting the ’s in objective function
(33) gives optimal upper bound on COST (2.5921 million $) of
.
supply chain with
Fortunately, in the present situation the global minimum point
becomes a design point so we need not proceed for any further
calculation. But if it is not so, then we will be required to solve
the underlying optimization problem by the Lagrange multiplier
method and get stationary points which satisfy the necessary
conditions. This is explained below.
Method of Lagrange Multipliers:
Lagrange Function: The Lagrange function
is given as
GARG et al.: DESIGN OF SIX SIGMA SUPPLY CHAINS
Fig. 11.
55
Optimal inventory level R for DP = 5 and DS = 0:9.
where
and is given by
(33).
Necessary Condition for Stationary Points: Let point
correspond to a local optimal point,
then this point must satisfy the following necessary conditions
for being a stationary point:
These necessary conditions result in the following relations:
Solving the above system of equations, by some numerical technique, will give the desired stationary points. First of all those
stationary points are discarded which are either imaginary or for
which the nonnegativity condition does not hold. After this, we
apply second order conditions to determine whether the point is
a maxima or a minima. Among all the minima points, the one
which yields minimum COST is considered as the solution of
the problem and we call it as the optimal upper bound on COST
yielding at least
, and
.
for
G. Some Numerical Results
Note that in the last section, we studied an instance of the
, and
for
IOPT problem assuming
. We obtained the optimal allocation of standard deviations
to achieve a minimum COST. The standard deviations obtained
can be used by a supply-chain manager to decide among alternate logistics providers or alternate suppliers, etc.
To obtain the optimal value of , we repeat the solution of the
IOPT problem for different values of , each time computing
the optimal upper bound on COST and the corresponding allocation of variabilities. The results of the above problem are
summarized in the following four plots, one for each set of constraints. Each curve represents the variation of optimal upper
bound on COST ($/year) with inventory level .
One can observe the following trends.
1) The optimal value of inventory is
for the first set
of constraints. This optimal value changes for a different
set of constraints.
increases as the desired quality level
2) The optimal level
increases. This can be verified by observing the trends
of the plots. Notice that for lower values of DP and DS,
keeping inventory is always profitable. If desired DP and
DS levels are high, the variabilities of the individual processes must be low enough to afford the luxury of having
very less inventory. It results in higher cost. In such a situation, higher inventory levels can only allow us to have
luxury of high DP and DS.
turns out to be equal to zero.
3) We found in some cases
This means that an inventory less system may be the best
option under some circumstances, providing a major case
for zero inventories.
56
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 1, NO. 1, JULY 2004
Fig. 12. Optimal inventory level R for DP = 6 and DS = 1:0.
V. SUMMARY AND FUTURE WORK
In this paper, we have presented a novel approach to achieve
variability reduction, synchronization, and therefore delivery
performance improvement in supply-chain networks. Our
approach exploits connections between design tolerancing
in mechanical assemblies and lead time compression in
supply-chain networks. The contributions of this paper can be
summarized as follows.
• Defining the notion of six sigma supply chains, to describe
supply-chain networks with superior levels of DS and DP.
• Describing design of six sigma supply chains as a
mathematical programming problem, thus providing a
sound framework for studying a rich variety of design
optimization and tactical decision making problems in
supply chains
• Illustrating the efficacy of the approach by formulating
and solving the IOPT problem for a four-stage LPG
supply-chain network.
The paper leaves plenty of room for further work in several
directions. The design problem that we studied here is only
one of a rich variety of design optimization problems that one
can address in the framework developed in this paper. Many
other problems, as listed in Section III.F can be studied. Also,
the supply-chain example that we have looked at belongs to
the make-to-order type and has only one stock. Multiechelon
supply chains with multiple inventories and make-to-order
policy would be the next category of systems that can be
addressed using our methodology. There is no reason why our
approach cannot be applied for coordination types other than
make-to-order, such as make-to-stock (MTS) and build to order
(BTO).
We believe the concepts and approach developed in this paper
provide a framework in which a rich variety of supply-chain
design and tactical decision problems can be addressed. Developing a sound methodology for rigorous design of complex
supply-chain networks is the ultimate goal of this research.
ACKNOWLEDGMENT
The first two authors would like to deeply appreciate the
collaboration with the Manufacturing Systems Research
Laboratory, GM R & D, Warren, MI. In particular, we thank
Dr. J. D. Tew, Dr. D. Kulkarni, and Dr. E. Foster of the above
laboratory for useful discussions.
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Systems. Englewood Cliffs, NJ: Prentice Hall, 1963.
D. Garg (S’02) received the M.Sc. (Eng.) degree in
computer science and automation, in 2002, from the
Indian Institute of Science, Bangalore, India, where
he is currently working toward the Ph.D. degree.
His current research interests include supply-chain
management, electronic commerce, mechanism design, and game theory.
Mr. Garg was awarded the Infosys Trophy
for Excellence in Management Research in
COSMAR-2002 and the Best Presentation Award
for Perspective Seminar Series held at Department
of Computer Science and Automation, Indian Institute of Science, Bangalore,
during January–April 2003.
57
Y. Narahari (SM’02) received the M.E. and Ph.D.
degrees in computer science from the Indian Institute of Science, Bangalore, India, in 1984 and 1988,
respectively.
He is currently a Professor in the Department
of Computer Science and Automation, Indian
Institute of Science. His current research interests
are in electronic commerce, auction and mechanism
design, dynamic pricing models, supply-chain
design and optimization, and software architecture
for e-business. In 1992, he was a Visiting Researcher
on sabbatical at the Massachusetts Institute of Technology, Cambridge, and
in 1997, at the National Institute of Standards and Technology, Gaithersburg,
MD. He has consulted for several companies including Intel, General Motors
Research, WIPRO, HCL, Satyam, and Tektronix. He has coauthored a widely
acclaimed textbook Performance Modeling of Automated Manufacturing Systems, (Englewood Cliffs, NJ: Prentice-Hall, 1992) and has recently completed
a very popular web-based text on data structures and algorithms.
Prof. Narahari has been on the editorial board of IEEE TRANSACTIONS ON
ROBOTICS AND AUTOMATION, and is currently on the editorial board of IEEE
TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS. The awards secured by
Prof. Narahari include: the Indian Institute of Science Best Ph.D. Thesis Award
in 1988; the Indo-US Science and Technology Fellowship in 1992; and the Sir
C. V. Raman Young Scientist Award for Computer Science Research, in 1998.
N. Viswanadham (F’93) is Deputy Executive
Director of The Logistics Institute-Asia Pacific,
Singapore, and also a Professor in Department of
Mechanical and Production Engineering, National
University of Singapore (NUS), Singapore. He was
a Tata Chemicals Chair Professor at the Indian
Institute of Science, Bangalore, India. He has
contributed significantly to the area of automation,
and, in particular, manufacturing automation. He is
the author of three textbooks, six edited volumes,
over 75 journal articles, and more than a 100
conference papers on automation. He is the lead author of the textbook entitled
Performance Modeling of Automated Manufacturing Systems, (Englewood
Cliffs, NJ: Prentice-Hall, 1992). His recent book is entitled Analysis of
Manufacturing Enterprises—An approach to leveraging the value delivery
processes for competitive advantage (Norwell, MA: Kluwer, 2000). Apart from
his widely followed books, he has published a number of high-impact papers
on quantitative performance analysis of production, product development,
logistics and supply-chain processes of a manufacturing enterprise. His current
research interests include logistics, supply-chain networks, and manufacturing
systems, and he is developing a methodology for design of six-sigma, synchronized manufacturing and service supply-chain networks. His research has very
strong practical as well as theoretical flavors and impact.
Prof. Viswanadham was a GE Research Fellow during 1989–1990 and the recipient of the 1996 Indian Institute of Science Alumni award for excellence in research. He is an Editor of IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION
and the IEEE TRANSACTIONS ON AUTOMATION SCIENCES AND ENGINEERING.
He is also a Fellow of the Indian National Science Academy, Indian Academy
of Sciences, Indian National Academy of Engineering, and the Third World
Academy of Sciences